Definition 5.1.1.

A projection in a -algebra is said to be an infinite projection if it is Murray-von Neumann equivalent to a proper su projection of itself. In symbols, if such that . If is not an infinite projection then we say that is a finite projection. A unital -algebra is said to be a finite C-star algebra if its unit is a finite projection . Otherwise we say that is an infinite C-star algebra . If is a finite C-star algebra for all , then we say that is stably finite.
If is non-unital, we classify with this terminology after unitizations.

NOTE: A projection in a -algebra is a finite projection if and only if is a finite C-star algebra.